Suppose that $3$ is a factor of $a$, $a$ is a divisor of $12$, and $a$ is positive. What is the number of possible values of $a$?
The positive divisors of $12$ are $1, 2, 3, 4, 6,$ and $12.$ Of these factors, $3$ is a divisor of $3, 6,$ and $12.$ We have a total of $\boxed{3}$ possible values of $a.$